A Coulomb-related family of potentials having zero-energy bound states

•Central potentials are proposed.•Zero-energy bound states are found.•Point transformation is discussed.•Quantum–classical comparison is made.

We propose here a new large class of singular central potentials having zero-energy bound states for many values of angular momenta. The potentials are shown to be closely related to the standard attractive Coulomb interaction. Some of them admit the E≠0E≠0 bound states as well. A quantum–classical correspondence is also discussed.